Problem: Given $ \overrightarrow{OL}\perp\overrightarrow{ON}$, $ m \angle LOM = 2x + 34$, and $ m \angle MON = 2x - 16$, find $m\angle LOM$. $O$ $L$ $N$ $M$
Explanation: From the diagram, we see that together ${\angle LOM}$ and ${\angle MON}$ form ${\angle LON}$ , so $ {m\angle LOM} + {m\angle MON} = {m\angle LON}$ Since we are given that $\overrightarrow{OL}\perp\overrightarrow{ON}$ , we know ${m\angle LON = 90}$ Substitute in the expressions that were given for each measure: $ {2x + 34} + {2x - 16} = {90}$ Combine like terms: $ 4x + 18 = 90$ Subtract $18$ from both sides: $ 4x = 72$ Divide both sides by $4$ to find $x$ $ x = 18$ Substitute $18$ for $x$ in the expression that was given for $m\angle LOM$ $ m\angle LOM = 2({18}) + 34$ Simplify: $ {m\angle LOM = 36 + 34}$ So ${m\angle LOM = 70}$.